In this paper, we have analytically studied rogue wave patterns in the NLS equation. We have shown that when one of the internal parameters in the bilinear rogue wave solutions is large, these waves would exhibit clear geometric structures, which comprise Peregrine rogue waves organized in shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at its center. These rogue patterns are analytically determined by the root structures of the Yablonskii-Vorob'ev polynomial hierarchy, and their orientations are controlled by the phase of the large parameter. We have also generalized these results and shown that, when multiple internal parameters in the rogue waves are large but satisfy certain constraints [such as (REF ) and (REF )], then the same rogue patterns would persist. Comparison between true rogue patterns and our analytical predictions has shown excellent agreement. As a small application of our analytical results, the numerical observation in [1]} on single-shell ring structures has been explained. Our results reveal the deep connection between NLS rogue wave patterns and the Yablonskii-Vorob'ev polynomial hierarchy, and make prediction of sophisticated patterns in higher-order NLS rogue waves possible.
